Arch. Mech., 62, 1, pp. 21–48, Warszawa 2010
Viscoelasticity and fractal structure in a model of human lungs
C. M. IONESCU1) , W. KOSIŃSKI2) , R. DE KEYSER1)
1)
Department of Electrical Energy, Systems and Automation
Ghent University
Technologiepark 913, Gent 9052, Belgium
e-mails: [email protected], [email protected]
2)
Department of Computer Science
Polish-Japanese Institute of Information Technology
Koszykowa 86, 02-008 Warszawa, Poland
and
Institute of Mechanics and Applied Computer Science
Kazimierz Wielki University in Bydgoszcz
Chodkiewicza 30, 85-064 Bydgoszcz, Poland
e-mail: [email protected]
This paper provides a model of the human respiratory system by taking into
account the fractal structure of the airways and the viscoelastic properties of
the tissue. The self-similarity of airway distribution is admitted up to the 24th
generation. Due to periodic breathing which results in sinusoidal excitation of the
respiratory system, an electrical equivalent model is developed. The periodic current
in this electrical network, that preserves the geometry of the human respiratory tree,
is equivalent to the oscillatory air-flow. The model is expressed by Navier–Stokes
equations under cylindrical symmetry, linked with an equation responsible for the
motion of viscoelastic tissue of airway walls. By use of both electro-mechanical
analogies, the total impedance of the respiratory system is determined and compared
to the measured data in the clinical range of 4–48 Hz, as well as in the low-frequency
range of 0.1–5 Hz. We propose also a lumped model of fractional orders, which is
able to capture frequency-dependent variations in both clinical as well as in the
low-frequency ranges. The models proposed in this paper can be further used to
determine the effects of disease on the lung morphology.
Key words: respiratory system, airways, Navier–Stokes flow, viscoelastic properties,
morphology, electrical transmission lines, input impedance.
c 2010 by IPPT PAN
Copyright °
Notations
p
δ Womersley parameter = R ωρ/µ,
έ10 , έ´10 phase angle of the complex number from Bessel functions of rank 1 and order 0,
respectively 1,
φ, ϕ phase angle,
γ complex propagation coefficient,
κ cartilage fraction,
22
C. M. Ionescu, W. Kosiński, R. De Keyser
λ
ψ
µ
νP
θ
ρ, ρwall , ρs , ρc
ω
ζ
cx
c̃, ć0 , c0
wavelength,
damping factor,
dynamic viscosity,
coefficient of Poisson (= 0.45),
contour coordinate,
density of air at BTPS, respectively of the airway wall of the soft tissue and of
the cartilage,
angular frequency,
radial deformation,
capacity per distance unit,
the complex velocity of wave propagation, the effective/corrected Moens–
Korteweg velocity an the nominal Moens–Korteweg velocity,
frequency in Hz,
conductance per distance unit,
wall thickness, √
complex unit = −1,
inductance per distance unit,
airway depth,
pressure,
flow,
radial direction, radial coordinate,
resistance per distance unit,
time,
velocity in radial direction,
velocity in contour direction,
velocity in axial direction,
axial direction, longitudinal coordinate,
ratio of radial position to radius = r/R,
compliance,
complex elasticity modulus, respectively for soft and cartilage tissue,
conductance,
Bessel function,
airway length,
inertance,
modulus for pressure gradient,
f
gx
h
i
lx
m
p
q
r
rx
t
u
v
w
z
y
Ce
E ∗ , Es , Ec
Ge
J
`
Le
M
´
Ḿ10 , Ḿ10 modulus of the complex number from Bessel functions of rank 1 and order 0,
respectively 1,
P pressure,
q flow,
Re resistance,
R airway inner radius,
Zl , Zt longitudinal, respectively transversal impedance.
1. Introduction
Lung geometry and morphology have been studied in the past from lung
casts and nowadays they have been validated using CT scans in 3D form [31, 32].
Already since Weibel in 1963 [39], the fractal geometry present in the lung morphology has been employed in studies on lung aerodynamics. It is significant to
Viscoelasticity and fractal structure in a model . . .
23
note that the self-similarity is related to the optimality of ventilation and that
asymmetry exists in the healthy lung as well, whereas a diseased lung contains
significant heterogeneities and the optimality conditions are not fulfilled anymore [12]. The ultimate goal of this study is to relate human lung morphology
to the properties of dynamic systems posing a fractal geometrical structure.
One of the most comprehensive and earliest overviews on the mechanical
properties of lungs is given by Mead in [22], describing the initial attempts to
quantify static and dynamic resistive, inertial and compliant properties of lungs.
His review covers both the inspiratory and expiratory phase, at laminar and
turbulent flow conditions, in terms of a single variable: the air volume. Another
important study has been reported in [26] for tube-entrance flow and pressure
drop during inspiration under spontaneous ventilation conditions. In the study
performed by Pedley in [29] to assess the flow and pressure drop in branching
airways, an important finding is that during spontaneous ventilation (unforced
breathing in relaxed conditions, also referred to as tidal breathing), the air flow
remains laminar (typical Reynolds number below 2000) [12, 26, 29].
Based on the technological and the computational progress, Sauret et al.
performed a study based on CT scans of the 3D-topology and morphology of
a human (cast) lung [31, 32]. Mean gravity and branching angles up to level 9
generations for the right and left lobe (asymmetric morphology due to heart
location) were reported. The present model differs from the previously reported
models in that it introduces the assumption of flexible airways in a simple form,
taking into account the wall tissue structure (in terms of cartilage and soft tissue
percent and densities) and derives the mechanical parameters to directly illustrate changes in airway morphology with disease. The resulting model is therefore
relatively simple and efficient to capture the intrinsic viscoelastic properties in an
electrical equivalent. There are several assumptions which are generally accepted
from previous studies and used here as simplifying assumptions [9, 22]. The first
assumption is that the pressure at the boundaries of all parts is the same at all
points of the respective boundaries. The next: three of the boundaries contain
gas only on one side: airway opening, alveolar surface and body surface. Uniform
pressure is valid if the gas is in continuity condition and no flow. These conditions are fulfilled for the body surface, during panting at the airway opening
and the alveoli. The only part which is not in agreement with this hypothesis is
the pleural surface, which has tissue on both sides and its pressure distribution
cannot be predicted. The work presented in this paper is based on laminar flow
conditions [12, 26, 29], which are indeed valid during tidal breathing.
The contribution of this work consists in deriving a model which combines
tidal breathing conditions and lung morphology. Although the theoretical basis
has been previously employed in modelling of the circulatory system under similar assumptions [27], the model proposed here allows to make variations in the
24
C. M. Ionescu, W. Kosiński, R. De Keyser
lung’s morphology as expected in specific pathologies. Making use of the fractal geometry of the respiratory tree and theoretical development for modelling
the circulatory tree, a mathematical model has been developed to characterize
the dynamics of respiratory pressure and flow [17]. The electrical analogy for
viscoelastic airways proposed here, provides insight into changes in resistance,
inertance, compliance and conductance parameters in healthy and diseased airways, where rheological properties play an important role. The procedure used
here allows to determine the correspondence between the electrical and mechanical parameters such as resistance, inertance, compliance and conductance of the
respiratory input impedance.
The original contribution of this work is the introduction of viscous loses into
the mechano-electrical equivalent model representation. It is important to quantify this aspect since the heterogeneity of the lung parenchyma is increasing
with pathology, and hence viscous loses become important. The previous model
consisting solely of elastic components will not be able to capture the changes
in the respiratory impedance with disease [17]. The work of breathing, i.e. the
energy necessary to be posed by the system to perform the respiratory function
is increasing, if there is a more viscous, stiff, scarred lung tissue (e.g. fibrosis,
emphysema).
The paper is further divided into four sections: the model for the pressure
and flow in the airways considered with viscoelastic wall properties, with specific morphologic parameters, the electrical equivalent representation, the results
of these models and discussion. A conclusion section summarizes the outcome
of this work and gives a short summary of its prospective use.
2. Materials and methods
2.1. Mathematical modelling of pressure-flow dynamics
For the case when sinusoidal excitation is applied to the respiratory system,
one may analyze the oscillatory flow conditions [15, 28, 9]. To find an electrical
equivalent of the respiratory duct, one needs the expressions relating pressure and
flow to properties of the viscoelastic tubes, which can be done straightforward via
the Womersley theory [2, 27]. Analogue to the Womersley theory from circulatory
system analysis, which considers the pulsatile flow in a circular pipeline, for
varying pressure-gradient, the periodic breathing (usually, for normal breathing
conditions, around 4 s) can be defined as a function of modulus and phase,
i.e. as a periodical function. Since the input to the respiratory system is the
pressure gradient, first we have to give the pressure gradient at z = 0 in terms
of a periodical function
∂p
= Re[M ei(ωt−φ) ] = M cos(ωt − φ),
(2.1)
−
∂z
Viscoelasticity and fractal structure in a model . . .
25
√
where z is the axial direction, longitudinal coordinate, i = −1, ω = 2πf is the
angular frequency (rad/s), with f the frequency (Hz), M the modulus and φ is
the phase angle of the pressure gradient. Given its periodicity, it follows that
the pressure and other velocity components will also be periodic, with the same
angular frequency ω, the velocity in radial direction u(r, z, t), the velocity in
the axial direction w(r, z, t), the pressure p(r, z, t), the dimensionless parameter
y = r/R, 0 ≤ y ≤ 1 denoting the ratio of the radial position with respect to
the axis of the tube are employed, along with all variables from Table 1 that
represents the tube parameters of each generation.
Table 1. The tube parameters for the sub-glottal airways generations, whereas 1
denotes the trachea and 24 the alveoli, as from [12, 13, 20, 23, 39, 40].
Generation
m
Length
` (cm)
Radius
R (cm)
Wall thickness
h (cm)
Cartilage
fraction κ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
10.0
5.0
2.2
1.1
1.05
1.13
1.13
0.97
1.08
0.950
0.860
0.990
0.800
0.920
0.820
0.810
0.770
0.640
0.630
0.517
0.480
0.420
0.360
0.310
0.80
0.6
0.55
0.40
0.365
0.295
0.295
0.270
0.215
0.175
0.175
0.155
0.145
0.140
0.135
0.125
0.120
0.109
0.100
0.090
0.080
0.070
0.055
0.048
0.3724
0.1735
0.1348
0.0528
0.0409
0.0182
0.0182
0.0168
0.0137
0.0114
0.0114
0.0103
0.0097
0.0094
0.0091
0.0086
0.0083
0.0077
0.0072
0.0066
0.0060
0.0055
0.0047
0.0043
0.67
0.5000
0.5000
0.3300
0.2500
0.2000
0.0922
0.0848
0.0669
0.0525
0.0525
0.0449
0.0409
0.0389
0.0369
0.0329
0.0308
0.0262
0.0224
0.0000
0.0000
0.0000
0.0000
0.0000
The air in the airways is treated as Newtonian, with constant viscosity µ =
1.8·10−5 kg/m-s and density ρ = 1.075 kg/m3 . Applying Navier–Stokes equations
in cylindrical coordinates [41], under the assumption of axi-symmetrical flow in
a cylindrical pipeline, air in the lungs is incompressible, and with vanishing
external body forces, we have:
26
(2.2)
C. M. Ionescu, W. Kosiński, R. De Keyser
µ
¶
·
µ
¶
¸
∂u
∂u
∂u
∂p
1 ∂
∂u
u
1 ∂2u ∂2u
ρ
+u
+w
=−
+µ
r
− 2+ 2 2 − 2
∂t
∂r
∂z
∂r
r ∂r
∂r
r
r ∂θ
∂z
for the radial direction r, and
µ
¶
·
µ
¶
¸
∂w
1 ∂
∂w
∂w
∂p
∂w
∂2w
(2.3)
ρ
+u
+w
=−
+µ
r
+
∂t
∂r
∂z
∂z
r ∂r
∂r
∂z 2
in the axial direction z, and
(2.4)
u ∂u ∂w
+
+
=0
r
∂r
∂z
as the continuity (incompressibility) equation. Dividing by the density ρ, observing that air in airways has very low total pressure drop variations, ≈ 0.1 kPa
[26], and using the relation y = r/R to substitute the partial derivative with
respect to r by ∂/∂y , and considering the further simplifying assumptions:
(i) the radial velocity component is small, as well as the ratio u/R and the
term in the radial direction;
(ii) the terms ∂ 2 /∂z 2 in the axial direction are negligible.
The following system is reached:
·
¸
∂u
1 ∂p µ 1 ∂u
1 ∂2u
u
=−
+
+
(2.5)
− 2 2 ,
∂t
ρR ∂y
ρ yR2 ∂y R2 ∂y 2
R y
·
¸
1 ∂p µ 1 ∂w
1 ∂2w
∂w
=−
+
+
(2.6)
,
∂t
ρ ∂z
ρ yR2 ∂y
R2 ∂y 2
(2.7)
1 ∂u ∂w
u
+
+
= 0.
Ry R ∂y
∂z
Studies on the respiratory system using similar simplifying conditions can be
found in [9, 26, 29]. Given the periodicity of the pressure gradient in (2.1),
it follows that also the pressure p(y, z, t) and the other velocity components
u(y, z, t), w(y, z, t) are periodic in time and z variable, as in:
p(y, z, t) = p1 (y)eiω(t−z/c̃) ,
(2.8)
u(y, z, t) = u1 (y)eiω(t−z/c̃) ,
w(y, z, t) = w1 (y)eiω(t−z/c̃) ,
where c̃ denotes the complex velocity of wave propagation and for the solution,
the real parts of the right-handed sides of (2.8) should be taken. Introducing in
(2.6) the solution given for p and w in (2.8), one obtains the following differential
equation:
(2.9)
d2 w1 (y) dw1 (y) iωρR2
iωρR2
+
−
w
(y)
=
−
p1 (y).
1
dy 2
ydy
µ
µρc̃
Viscoelasticity and fractal structure in a model . . .
27
Its solution is given by
w1 (y) = C1 J0 (λy) +
iωR2
p1 (y),
µc̃(λ2 − k 2 )
with C1 being a complex integration constant, J0 a Bessel function of the 1st
kind and 0 degree, and k an arbitrary constant which is to determine via the
continuity equation (2.7) and λ2 = i3 δ 2 with
p the Womersley parameter δ, defined
as the dimensionless parameter δ = R ωρ/µ [17, 43]. The further detailed
procedure of finding the solutions for p1 and u1 is described in [34] where all
equations (2.5)–(2.7) are used. It turns out that the constant k = iωR/c̃ is
a measure of proportionality of the tube radius to the wavelength of a pressure
wave (c̃/2πf ), and resulting in small values. For the tracheal respiratory tube
where R = 0.008m and for the breathing frequency of 0.2–5 Hz, the wavelength
is about 2.5 m. Further simplifications lead to the following forms of solution:
½
¾
iωR
2
A
(2.10)
u(y, z, t) =
C1 J1 (λy) +
y eiω(t−z/c̃) ,
µc̃
λ
ρc̃
¾
½
A iω(t−z/c̃)
(2.11)
w(y, z, t) = C1 J0 (λy) +
e
,
ρc̃
∂p
= M ei(ωt−φ) ,
∂z
with C1 = −A/J0 (λ)ρc̃, J1 a Bessel function of rank 1 and the 1st degree, and
(2.12)
p(1, z, t) = Aeiω(t−z/c̃)
−
or
−
iω
∂p
= Aeiω(t−z/c̃) = M ei(ωt−φ) .
∂z
c̃
c̃
Hence due to (2.1), the compatibility condition is Aeiω(t−z/c̃) = M ei(ωt−φ−π/2) .
ω
The coefficients A and M will be determined from the interface and boundary
conditions.
2.2. Viscoelasticity of airways wall
The boundary condition linking the wall and pipeline equations is the no-slip
condition that assumes the fluid particles adherent to the inner surface of the
airway, and hence to the motion of the viscoelastic wall.
The viscoelasticity of the wall is determined by the rheological properties of
the tissue. Hence, it depends on the amount of cartilage fraction in the tissue, as
the viscous component (collagen), or by the soft tissue fraction in the tissue as
the elastic component (elastin) [6]. Taking into account at each level the fraction
amount κ of the corresponding cartilage tissue (index c) and soft tissue (index s)
and with Ec = 400 kPa, Es = 60 kPa, ρc = 1140 kg/m3 , ρs = 1060 kg/m3 , we
have that the effective wall tissue density is:
28
(2.13)
C. M. Ionescu, W. Kosiński, R. De Keyser
ρwall = κρc + (1 − κ)ρs .
In the previous elastic case [17], Hooke’s law was employed, i.e. the stress is
related to the strain by the linear relation:
(2.14)
σ = E²
with the effective elastic modulus calculated as: E = κEc + (1 − κ)Es for each
airway generation. However, this is not valid in case of viscoelastic airway walls,
as shown in [35]. In the present study, the strain measure is given by ² = ζ/R and
the uniaxial stress σ has been substituted by its fraction σD = σ/(1 − νP2 ). The
equivalent of (2.14) is the ratio between stress and strain of the lung parenchymal
tissue with the Young’s moduli E.
To incorporate the new constitutive equation in the present model of human
lungs, let us come back to our previous model and calculation from [17]. In our
sequel, the modelling of the motion of the relatively short airway ducts has been
limited to radial movement of the tube. Hence its motion has been described
by the function ζ(z, t) of the axial variable z and the time t. For the strain
measure ², the ratio ζ/R where R is the initial radius of the tube, has been
adopted and the corresponding strain rate measure will be given by:
1 ∂ζ
∂²
=
.
∂t
R ∂t
(2.15)
In the case of viscoelastic model of the airway wall, the Voigt body model [6, 8]
can be asumed. The Voigt body is the simplest viscoelastic model that can
store and dissipate energy. It consists of a perfectly elastic element, i.e. a spring,
arranged in parallel with a purely viscoelastic element, i.e. a dashpot. Its constitutive equation for the stress σ in terms of the strain ² is:
(2.16)
σ = E² + η
∂²
,
∂t
where E represents the elastic constant of the string and η is the viscous coefficient of the dashpot. This observation together with (2.16)–(2.15), leads to the
following expression for the uniaxial stress in the wall of the tube:
(2.17)
σD =
η
E ζ
∂ζ
+
.
2
2
1 − νP R R(1 − νP ) ∂t
2.3. A fractional order lumped model of viscoelasticity
The viscoelasticity of the wall determined by the rheological properties of the
tissue and described by (2.16) does not take into account the stress relaxation
Viscoelasticity and fractal structure in a model . . .
29
effect, because in the process under constant strain ²(t) = ²(t0 ) for t > t0 ,
the stress does not change, since then the strain rate is zero and consequently
σ(t) = E²(t) = E²(t0 ). Taking the time derivative of (2.16), we get:
(2.18)
∂σ(t)
∂²
∂ 2 ²(t)
=E +η
.
∂t
∂t
∂t2
This equation does not show relaxation either. Let us add to the right-hand side
of the last equation an extra term proportional to the strain, i.e. H², with the
new material parameter H. We obtain a new constitutive law:
(2.19)
∂σ(t)
∂²
∂ 2 ²(t)
+ H²(t).
=E +η
∂t
∂t
∂t2
Notice that this constitutive equation takes into account relaxation, because
in the process under constant strain ²(t) = ²(t0 ) for t > t0 , the stress will change.
Since the strain rate is zero, the extra term will give the stress rate:
(2.20)
∂σ(t)
= H²(t0 ),
∂t
and consequently σ(t) = σ(t0 ) + (t − t0 )H²(t0 ). Notice that in the relaxation
process we are expecting a decay of the stress to zero, which takes place only if
H is negative. Relation (2.19) is a mixture of the Maxwell and Voigt models of
linear viscoelasticity.
Finally, both the Maxwell-element, as well as the Kelvin–Voigt-element, do
not fully characterize the true viscoelastic behaviour. Hence, combining of both
elements seems to be a good solution to overcome their individual limitation:
N parallel Maxwell-elements, all in parallel with an extra spring, as shown
in Fig. 1.
Fig. 1. From left to right: the Maxwell- and the Kelvin–Voigt-element, followed by a
combination of the two.
30
C. M. Ionescu, W. Kosiński, R. De Keyser
For linear 1D viscoelastic material of Voigt type the classical Young modulus,
denoted by E, after the Fourier transform of (2.16) does not get any frequency
term as a product, due to the relation
(2.21)
EV∗ (iω) =
σ̂(ω)
= E + ηiω.
²̂(ω)
If we apply the Fourier transform to the the second law of the mixture type (2.19),
we get
(2.22)
∗
EM
(iω) =
σ̂(ω)
= E + i(ηω − H/ω) ,
²̂(ω)
∗ on frewhich means a nonlinear dependence of complex viscoelastic modulus EM
quency. The question arises whether this nonlinear relation is in accordance with
clinic observations or how to model different type relations. To answer this question, we refer to the concept of fractional order calculus (FOC) integral in characterizing the viscoelastic properties of the arterial walls modeled by Craiem
and Armentano [8]. Clinic observations have shown that complex interactions
in the lung tissue are significant at low frequencies [4], i.e. close to the breathing
frequency, making unbiased identification a very difficult task. Moreover, elastin
plays an important role in determining the rheological behavior of soft tissue,
which in turn leads to the appearance of FOC differ-integration [36, 37]. The
viscoelastic properties of lung tissue cause the effective tissue resistance to be
very high at very low frequencies, but then to decrease asymptotically towards
zero, at higher frequencies. Consequently, the tissue resistance has the main contribution to the total lung resistance. At breathing frequencies in the range of
0.2–0.4 Hz, tissue resistance can account for ≈ 40% of the lung resistance.
Our previous observations show that in case of a whole model of the respiratory system with input impedance, both the integral and differential FOC
must be present. Due to the complex nonlinear phenomena exhibited in the
lung parenchyma and its intrinsic viscoelastic nature, it is difficult to relate
the mechanical properties of the lungs to integer order differential system, like
the mixed Maxwell–Voigt model (2.19). It is clear that fractional calculus offers handy tools to characterize such complex dynamics. It is also challenging to
explain viscoelasticity in relation to lung pathology.
Fractional order calculus stems from the beginning of the theory of differential and integral calculus [24, 25, 30] and has been a fruitful field of research
in science and engineering. In seems that in the last two decades, fractional
differentiation has played an increasing role in various fields such as mechanics
(viscoelasticity/damping,), electricity, electronics, chemistry, biology, economics
and notably control theory, robotics, image and signal processing, diffusion and
wave propagation [10, 33, 38].
Viscoelasticity and fractal structure in a model . . .
31
Another and very broad field of recent applications of FOC is biomechanics.
Some diseases (such as osteoporosis) are caused by biochemical and hormonal
changes in human body. They lead to modification of the structure and composition of bones such as porosity and thickness of trabeculae, as well as to mineral
density changes. Since trabecular bone is an inhomogeneous porous medium with
viscoelastic properties, to measure those changes ultrasonic methods are used.
The interaction between the ultrasound and the bone is highly complex. Modelling of ultrasonic propagation through trabecular tissue has been considered by
using porous media theories, e.g. Biot’s theory. Many authors have used fractional calculus as an empirical method to describe the properties of porous and
viscoelastic materials [10, 33]. We are following this line-of-thought in modelling
of the lung tissue.
There are two different viewpoints to the fractional calculus: the continuoustime viewpoint based on the Riemann–Liouville fractional integral [25] and
the discrete-time viewpoint based on the Grünwald–Letnikov fractional derivative [30]. Both approaches turn out to be useful in treating situations of practical
application in different fields, including numerical analysis, physics, engineering,
biology, economics and finance.
Take any real-valued function h(t) defined on R+ = [0, ∞); by a fractional
derivative of order α ∈ R of h we understand a function defined on [0, ∞) =: R+
with its value in R, given by the classical Caputo definition [30] as
(2.23)
dα h(t)
1
=
α
dt
Γ (n − α)
Zt
0
h(n) (τ )
dτ,
(t − τ )α+1−n
for n − 1 < α ≤ n. If α = n then
(2.24)
dα h(t)
= h(n) (t);
dtα
here h(n) (τ ) denotes the nth-order derivative of the function h(τ ), τ ∈ R+ and
Γ is the Euler gamma function, which generalizes the factorial to non-integer
values1) , i.e. Γ (x + 1) = xΓ (x) when x > 0.
Integer order derivatives are local operators, fractional, i.e. real order derivatives, on the other hand, are non-local ones. Following (2.23) they can be seen
as the convolution of h(t) with a tn−1−α function, anticipating some memory capability and power-law responses. In FOC negative order derivatives correspond
to integration, and the following formula holds which is an equivalence of the
1)
The function is extended to negative values by the definition Γ (x) = Γ (x + 1)/x, then
Γ (−1/2) = −2Γ (1/2). An extension to complex argument also exists.
32
C. M. Ionescu, W. Kosiński, R. De Keyser
Fourier transformation of integer order derivatives
(2.25)
α h(t)
d\
= (iω)α ĥ(ω).
dtα
If we pass, like Craiem and Armentano [8], to the fractional order derivatives in the last Equation (2.19), with different orders α and β for strain and
stress, respectively, then we get
(2.26)
∂ β σ(t)
∂β ²
∂ α+β ²(t)
=
E
+
η
+ H²(t),
∂tβ
∂tβ
∂tα+β
where 0 ≤ α, β ≤ 1. If we take the integral of fractional order β of the both sides
of (2.26), we obtain:
(2.27)
σ(t) = E²(t) + η
∂ −β ²(t)
∂ α ²(t)
+
H
.
∂tα
∂t−β
Taking the Fourier transform of the last relation we get:
(2.28)
σ̂(ω) = (E + η(iω)α + H(iω)−β )²̂(ω).
Now we can pass to our model of respiratory system with periodic breathing
condition (2.1), (2.8). Then it is helpful to introduce the complex value material
(pseudoelastic) modulus E ∗ which is a composition of both the material coefficients E and η and the function of frequency ω appearing on the RHD of (2.28)
in front of ²̂, namely:
(2.29)
where
E ∗ = Ere + iEim ,
µ ¶
µ
¶
πα
−πβ
α
Ere = E + η cos
ω + H cos
ω −β ,
2
2
µ ¶
µ
¶
πα
−πβ
α
Eim = η sin
ω + H sin
ω −β .
2
2
Its form will be used in the further calculation of the complex modulus of elasticity. Notice that the case of the classical Voigt material is included here, under
the condition: α = 1, β = 0.
2.4. Balance equation for viscoelastic thick-walled tube
To write the balance equation for an arbitrary element of the tube wall confined by the inner radius R and the outer radius R + h, the radial angles θ1 , θ2
and the boundaries in the axial directions z1 , z2 , we assume, that on the tube
at the inner radius R only the air pressure P acts, while the outer radius is
Viscoelasticity and fractal structure in a model . . .
33
stress-free [17]. Hence the equation of motion of the infinitesimal element dθdz,
where dθ = θ2 − θ1 and dz = z2 − z1 with thickness h follows, as in [17], from
the second law of motion which states the balance of gravitational, inner and
inertial forces. The gravitational forces are neglected and the inner forces are
the pressure p and the stress σD . Knowing that the inertial force is the effective
mass density ρwall times the volume and the acceleration, it follows that:
∂2ζ
,
∂t2
where σD is given by (2.17). Notice that (2.30) is valid for any increment dθdz.
Assuming a negligible displacement ζ in comparison to R, we may divide both
sides of (2.30) by Rdθdz, to get:
(2.30)
p(R + ζ)dθdz + hσD dθdz = hρwall (R + ζ)dθdz
σD
∂2ζ
= hρwall 2 .
R
∂t
Now incorporating the constitutive equation (2.17) for the stress, we obtain that:
µ
¶
1
E ζ
η
∂ζ
∂2ζ
(2.32)
p+h
+
=
hρ
.
wall
R 1 − νP2 R R(1 − νP2 ) ∂t
∂t2
(2.31)
p+h
If we arrange the appropriate terms, then we have:
¶
µ
∂2ζ
∂ζ
h
=
hρ
Eζ
+
η
.
(2.33)
p+ 2
wall
∂t
∂t2
R (1 − νP2 )
Here we can use a more general constitutive law describing viscous effects
given by fractional order derivative (2.26) or (2.27). Then we get
µ
¶
h
∂ α ζ(t)
∂2ζ
∂ −β ζ(t)
(2.34)
p+ 2
Eζ
+
η
+
H
=
hρ
.
wall
∂tα
∂t−β
∂t2
R (1 − νP2 )
From this one can obtain new relation for a given oscillatory frequency (or
an interval of frequencies).
Let us now summarize the model developed hitherto. The model for wave
propagation as a function of the pressure p (kPa), for axial w (m/s) and radial
u (m/s) velocities, for flow q (l/s), and for the wall deformation ζ at the axial
distance z = 0, is based on our previous studies and is given by the equations [17]:
(2.35)
(2.36)
(2.37)
p(t) = Aei(ωt−φP ) ,
¶
µ
´
RAω Ḿ10 (y)
π
´
·
,
u(y, t) =
cos ωt − έ10 − φP + έ10 (y) +
2
2ρć20
Ḿ10
µ
¶
R2 Aω
έ10
π
Ḿ0 (y)
p
w(y, t) =
sin ωt −
− φP + έ0 (y) +
,
·
δ2
2
2
ć0 µ Ḿ10
34
(2.38)
C. M. Ionescu, W. Kosiński, R. De Keyser
µ
¶
πR4
Aω
έ10
π
Ḿ10
q(t) =
· p
· 2 sin ωt +
− φP +
,
µ
2
2
ć0 Ḿ10 δ
A
cos(ωt − φP ),
− ρwall hω 2
µ
¶
E∗ h
2 , and where:
with A = 2R
−
ρ
hω
wall
1 − ν 2 R2
p
(2.40)
ć0 = E ∗ h/(2ρR(1 − ν 2 )).
(2.39)
ζ(t) =
hE ∗
R2
´
We denote by Ḿ0 , Ḿ10 , Ḿ10 the moduli, and by έ0 , έ10 , έ´10 the phase angles of
Bessel functions of the first kind and order 0, respectively 1 [1], as in:
Ḿ0 (y)eiέ0 (y) = 1 − J0 (δi3/2 y)/J0 (δi3/2 ),
(2.41)
Ḿ10 eiέ10 = 1 − 2J1 (δi3/2 )/(J0 (δi3/2 )δi3/2 ),
´
´
Ḿ10 (y)eiέ0 (y) = 1 − 2J1 (δi3/2 y)/(J0 (δi3/2 )δi3/2 ).
Equations (2.35)–(2.39) represent the model developed in [17]. This model is
derived from the previous works of [26, 27, 29, 34], which are based on the
classical fluid dynamics theory and hydraulics [41]. The derivation of the model
is presented in detail in [34], applied to the cardiovascular system. In our sequel
[17] we have adapted this model to the conditions of the respiratory tree.
Fig. 2. Schematic representation of the bronchial tree: generations 1–16 transport gas and
17–24 provide gas exchange [40].
Viscoelasticity and fractal structure in a model . . .
35
We introduced viscoelasticity in (2.29), assuming that the viscoelastic behaviour of the airway walls was a complex function, yielding a real and an imaginary part [6, 8, 35]. Relation (2.29) can be then written as a corresponding
modulus and phase: E ∗ = Ere + iEim = |E|eiϕE . The complex definition of
elasticity will change the form of the wave velocity from (2.40) to:
s
s
iϕ
E
|E|he
|E|h
(2.42)
ć0 =
=
e0.5iϕE .
2ρR(1 − ν 2 )
2ρR(1 − ν 2 )
2.5. Electrical modelling of pressure-flow dynamics
By analogy to electrical networks [7], one can consider voltage P to be equivalent for respiratory pressure p and current Q as equivalent for air-flow q (see
Table 2). Electrical resistances Re represent respiratory resistance that occurs
as a result of airflow dissipation in the airways, electrical capacitors Ce represent volume compliance of the airways which allows them to inflate/deflate,
and electrical inductors Le represent inertia of the air [17]. Additionally to these
components, we introduce the viscous losses represented by equivalent electrical
conductance Ge . These properties are often clinically referred to as mechanical
properties: resistance, compliance, inertance and conductance; and this section
will describe them in function of airway morphology.
Table 2. The electromechanical analogy.
Electrical
Mechanical
Voltage [V]
Current [A]
Resistance Re [Ω]
Capacitance Ce [F]
Inductance Le [H]
Force [N]
Velocity [m/s]
Damping constant B [N · s/m]
Spring constant 1/K [m/N]
Mass M [kg]
Consider a transmission line cell as a 2-input/output system depicted in
Fig. 3. Using the relations sinh(γx) = (eγx − e−γx )/2, cosh(γx) = (eγx + e−γx )/2,
we can write the relationship between the input x = −` and the output x = 0
as:

" # 
· ¸
cosh(γ`)
Z
sinh(γ`)
0
P1
P2


= 1
,
(2.43)
sinh(γ`) cosh(γ`)
Q2
Q1
Z0
p
p
with γ =
(rx + iωlx )(gx + iωcx ) =
Z√
l /Zt being the propagation conp
stant, Z0 =
(rx + iωlx )/(gx + jωcx ) = Zl Zt – the characteristic impedance and Zl = rx + iωlx = γZ0 – the longitudinal impedance, respectively
36
C. M. Ionescu, W. Kosiński, R. De Keyser
Fig. 3. Schematic representation of the infinitesimal distance dx over the transmission line
and its parameters.
Zt = 1/(gx + jωcx ) = Z0 /γ – the transversal impedance. The relation for the
longitudinal impedance in function of aerodynamic variables is given by
Zl =
iωρ
πR2 Ḿ10
e−iέ10 =
µδ 2
µδ 2
π
πR4 Ḿ10
e−i( 2 −έ10 ) =
πR4 Ḿ10
[sin(έ10 ) + i cos(έ10 )],
respectively, in terms of transmission line parameters, the longitudinal impedance is given by Zl = rx + iωlx . By equivalence of the two relations we have
that the resistance per unit distance is:
(2.44)
It follows that ωlx =
rx =
µδ 2
πR4 Ḿ10
µδ 2
πR4 Ḿ1 0
sin(έ10 ).
r
cos(έ10 ) and recalling that δ = R
ωρ
, the inµ
ductance per unit distance is:
(2.45)
lx =
ρ cos(έ10 )
.
πR2 Ḿ10
In case of a viscoelastic pipeline, the characteristic impedance is given by
s
ϕ
έ
ρ
|E| h 1
1
−i( 10
+ 2E )
2
p
Z0 =
e
πR2 1 − νP2
2ρR Ḿ10
and the transversal impedance is given by
µ
¶
2πR3 (1 − νP2 )2 i( π −ϕE )
1
Z2
Zt =
= 0 = 1/ ω
e 2
,
gx + iωcx
Zl
|E| h
from where the conductance per unit distance can be extracted:
(2.46)
gx = ω
2πR3 (1 − νP2 )2
sin ϕE
|E| h
Viscoelasticity and fractal structure in a model . . .
37
and the capacitance per unit distance is given by:
(2.47)
cx =
2πR3 (1 − νP2 )2
cos ϕE .
|E| h
Thus, from the geometrical (R, h) and mechanical (E ∗ , νP ) characteristics of the
airway tube, and from the air properties (µ, ρ), one can express the rx , lx , gx and
cx parameters. In this way, the dynamic model can be expressed in an equivalent
transmission line defined by Eqs. (2.44)–(2.47). Since |γ| ¿ 1, we can estimate
that over the length ` of an airway tube (thus x = `), we have the corresponding
properties:
µδ 2
(2.48)
Re = rx ` = `
(2.49)
Le = lx ` = `
(2.50)
Ge = gx ` = `ω
(2.51)
Ce = cx ` = `
πR4 Ḿ10
sin(έ10 ),
ρ cos(έ10 )
,
πR2 Ḿ10
2πR3 (1 − νP2 )2
sin ϕE ,
|E| h
2πR3 (1 − νP2 )2
cos ϕE .
|E| h
Some more details are given in the Appendix.
2.6. Applications
The set of equations given by (2.35)–(2.39) can be used to investigate the
variations in tidal breathing pressure and flow waves caused by pathology in the
nominal function of the lung. Tidal breathing means that air goes into the lungs
the same way that it comes out; in other words, it characterizes the inhalation
and exhalation at rest (no forceful maneuvers, no exercise, etc.). In this paper, we
shall investigate only the normal (healthy) case. However, one may expect that
variations in pressure and flow patterns from specific diseases will have effect
on the mechanical parameters (2.48)–(2.51), changing the total values of the
respiratory input impedance. Due to the fact that the network is dichotomous
and symmetric, we can obtain the total mechanical impedance using the network
structure as in Fig. 4, with Bm and Km calculated with the electro-mechanical
equivalence from Table 2. Since the Kelvin–Voigt elements corresponding to one
level are in parallel, their transfer function Hm will be in series with the spring in
the level m − 1. The next corresponding transfer function is in parallel with the
damper in the level m − 1, as depicted schematically by Fig. 4. In this manner,
the total transfer function H(s) can be determined, starting at level 24.
38
C. M. Ionescu, W. Kosiński, R. De Keyser
Fig. 4. A schematic representation of how the mechanical impedance H(s) is calculated
from level 24 (red dashed box) by adding consequent levels (green box, blue dashed box, etc.)
up to level 16.
Hence, in this representation, the stress and strain properties can be evaluated using (2.16). The strain is increased in steps of 10%, from 10% to 100%.
Starting from level 24, one can then calculate the stress-strain curve at the input of each level. This then will give rheological information in the context that
all parenchymal levels are interconnected, hence in a combination of Maxwell–
Kelvin–Voigt elements. Notice that here we show the stress-strain only for the
respiratory parenchyma (levels 16–24).
2.7. Measurements in the 4–48 Hz frequency range
In order to compare the results obtained by the nominal simulator parameters with measured data from healthy patients, an averaged impedance has been
made from 25 healthy subject whose biometric characteristics are given in Table 3 [18]. These subjects have been tested for their lung function using the forced
Table 3. Biometric parameters of the healthy subjects. Values are
presented as mean ± SD, as from [18].
Healthy
Age (yrs)
Height (m)
Weight (kg)
26 ± 3
1.67 ± 0.04
64 ± 3.7
Viscoelasticity and fractal structure in a model . . .
39
oscillation technique [28]. From the non-invasive measurement of the air-pressure
and air-flow at the mouth, the complex impedance is obtained, and its equivalent
Bode and polar characteristics [15]. The range of frequencies where the data is
measured varies between 25 rad/s and 300 rad/s (i.e. from 4 to 48 Hz).
2.8. Measurements in the 0.1–5 Hz frequency range
Since viscous effects are visible at low frequencies, we need to validate our
model at low frequencies (i.e. below 5 Hz). For the purpose of this validation,
we use the data reported in the specialized literature. The first set consists of
impedance data in healthy subjects in the frequency range between 0.25 and
5.0 Hz [14]. The measurements were taken in normal breathing conditions, using the forced oscillation technique. The second set of impedance data consists of ventilated anesthetized patients during cardiac surgery between 0.2 and
2.0 Hz [3]. The two data sets were selected from the published papers and approximated using the corresponding figures.
3. Results
The variations with each generation for the mechanical parameters resistance,
iterance, capacitance and conductance are given in Fig. 5. This variation becomes
linear when plotted as a logarithmic function, whereas it becomes exponential
when plotted linearly.
−4
capacitance (l/kPa)
resistance (kPa s/l)
2
10
0
10
−2
10
0
5
10
15
20
25
10
−2
10
−3
10
−4
10
0
−6
10
−8
10
0
5
10
15
20
25
5
10
15
generation
20
25
−4
conductance (l/kPa)
inertance (kPa s²/l)
−1
10
5
10
15
generations
20
25
10
−6
10
0
Fig. 5. Variations in: resistance, iterance, capacitance and conductance for nominal
(continuous line), case.
Figure 6(left) depicts the evolution of the mechanical parameters in a single
tube at a certain level m, whereas Fig. 6(right) depicts their evolution in the
entire level. One may observe that the evolution in a single tube, in consecutive levels is quasi-linear for both parameters (Fig. 6(left)). However, since the
40
C. M. Ionescu, W. Kosiński, R. De Keyser
−3
x 10
−6
10
Damperm (Ns/m)
Damperm (Ns/m)
6
5
4
3
−8
10
−10
2
16
17
18
19
20
21
22
23
10
24
16
17
18
19
20
21
22
23
24
17
18
19
20
21
airway generation m
22
23
24
−3
x 10
−6
10
Springm (N/m)
Springm (N/m)
8
6
4
2
16
−8
10
−10
17
18
19
20
21
airway generation m
22
23
24
10
16
Fig. 6. Parameter evolution in singular tubes (left) and in the entire level (right),
for levels 16–24.
total parameter values from Fig. 6(right) depend on the total number of tubes
within each level, they change as an exponential decaying function. When represented in a logarithmic scale, one can observe a quasi-linear behavior, as in
Fig. 6(right).
The stress-strain curves are depicted in Fig. 7. As expected, the stress increases with the degree of elongation applied to the entire structure. The more
levels we have in our structure, the higher will be the values of the stress-strain
curve, due to higher amount of cartilage tissue (collagen).
3
10
2
stress (Pa)
10
1
10
0
10 −1
10
0
10
strain (−)
Fig. 7. The stress-strain curves. The vertical arrow denotes the evolution in each level, from
a ladder network model of the cell 24, to ladder network models with additional cells, until
all levels until 16 are included.
Viscoelasticity and fractal structure in a model . . .
41
0.4
0.3
0.2
mean
lower bound
upper bound
model
0.2
0.1
0 1
10
10
2
10
3
0.3
0.2
Imaginary Part (kPa s/l)
Imaginary Part (kPa s/l)
Real Part (kPa s/l)
One can observe the fitting of the model to the averaged measured impedance
in the 4–48 Hz frequency interval depicted in Fig. 8(left). Its equivalent polar
plot is given in Fig. 8(right).
mean
lower bound
upper bound
model
0.15
0.1
0.05
0.1
0
0
-0.1 1
10
2
10
Omega (rad/s)
10
-0.05
0.05
3
0.1
0.15
0.2
Real Part (kPa s/l)
0.25
0.3
Fig. 8. Measured impedance in healthy subjects as averaged values with upper and lower
bounds (left); model fitting (right).
Magnitude (dB)
The evolution with frequency of the mechanical impedance as per level or
as a total branching network is given in Fig. 9. One may observe that at low
frequencies the frequency dependence becomes significant.
−100
−150
−200 −5
10
0
Phase (°)
0
−50
−100 −5
10
5
10
0
10
omega (rad/s)
10
16
17
18
19
20
21
22
23
24
5
10
Fig. 9. Variations in the mechanical impedance with each airway generation level.
Evaluation of the lumped model Zid = Rid +1/(Cid sβ ) in the lower frequency
range is done on the data known from literature, Zr . Figure 10 depicts the
42
C. M. Ionescu, W. Kosiński, R. De Keyser
Magnitude (dB)
0
−10
−20 0
10
Phase (°)
10
Zr
Zid
1
10
2
10
0
−5
0
−60
−20
−65
−40
−60
−80 0
10
1
10
Frequency (rad/s)
2
10
Zr
Zid
5
−10 0
10
Phase (°)
Magnitude (dB)
10
1
10
2
10
−70
−75
−80 0
10
1
10
Frequency (rad/s)
2
10
Fig. 10. Impedance data Zr from literature [14] (left) and from [3] (right), and the fitted
lumped model Zid .
fitting of the model on the two data sets from literature. The model values
for Fig. 10(left) are: total resistance 0.1332 (kPa s/l), compliance 0.5274 (l/kPa)
and a fractional order of β=0.8095. The model values for Fig. 10(right) are: total
resistance 0.0488 (kPa s/l), compliance 0.3274 (l/kPa) and a fractional order of
β=0.8035.
4. Discussion
The results presented here are in agreement with previous studies and show
that predominant pressure drop occurs in the upper airways (first 5 bifurcation
levels) [26]. The results depend strongly on the airway wall viscoelasticity, which
is varying in disease. For typical flow rates during spontaneous breathing ranging
0.5–1 l/s, wall roughness is neglected since it has little effect in laminar flow
conditions and for low values of the Womersley parameter [19]. In the respiratory
system, the values of δ are always less than 1, varying from 0.0471 in alveoli to
0.785 in the trachea. For the circulatory system, these values become as high
as 24, causing deviations from the Poiseuille parabolic flow profile. Therefore,
we can conclude that in our case the parabolic flow profile remains constant.
Our results are obtained under the following assumptions:
• laminar flow for typical Reynolds number during quiet breathing is less
than 2000;
• ducts are long enough (this assumption is not true, thus neglecting the
initial-flow length effects);
• the air is homogeneous and Newtonian;
• the axial velocity component is zero at the airway wall;
• uniform cylindrical duct (valid as approximation);
Viscoelasticity and fractal structure in a model . . .
43
• for linearization we have assumed the following simplifications:
a) − ωR
c ¿ δ, which is true, for in respiration we have values between
3.5904e−5 and 2.1542e−6 ;
b) the air velocity is small compared to the wave velocity; this is valid for
most of the airways; i.e. in trachea there may be velocities as high as
10m/s, with a wave velocity of 339m/s;
c) the values for y vary between 0 → ±1 (rigid pipe), although in reality
it varies between 0 → ±(1 + ζ/R) (viscoelastic pipe);
d) the E ∗ modulus is dependent on the airway wall structure (cartilage
fraction);
• thin-walled ducts; for the healthy respiratory system, the ratio h/R varies
between 0.4625 in trachea, to 0.0896 in alveoli.
Due to the fact that we do not seek to obtain a precise/exact value of the
components but merely a qualitative value, we assume that a more complex
formulation may be correct and more realistic, but may improve little the overall
conclusions.
For frequencies below 100Hz the transmission line theory can be applied in
a simplified form, leading to the exact solution for pressure and flow changes in
normal breathing conditions. A similar study has been employed in [27], leading to the same formula for the compliance (2.47). Similarity exists between
the derivation of the input impedance in the respiratory tree in this study and
modeling of the smaller systemic arteries, since in both simulations the symmetric structure is employed, along with laminar flow conditions, incompressibility,
Newtonian fluid and the non-slip boundary conditions. The input impedance
is extended to a more general tree in [27], by adding the equation of crossing
a bifurcation based on a law on which the geometry changes over the junction.
Nevertheless, we may argue that our choice of choosing to model a completely
symmetric tree still reflects its essential behaviour.
Notice that in both impedance data, the models identified lower values in
magnitude (or equivalently in the real part of complex impedance) in anesthetized subjects, is done by skipping the upper airways and measuring directly
in the trachea (intubated patients). Since sedation provokes a relaxation of the
parenchyma from a viscoelastic point of view, the compliance is lower in anesthetized patients than in those subjects during tidal breathing. This means that
the lungs of the anesthetized patients are posing more resistance to changes in
pressure, resulting in lower flow values. Typically, most of these patients are
under artificial ventilation during surgery. The value of this fractional order in
both modelling approaches is constant between the two sets of data. Since this
order encompasses both the viscous and elastic elements, the fact that both sets
of data come from subjects with healthy lung parenchyma, the identified values
are similar.
44
C. M. Ionescu, W. Kosiński, R. De Keyser
It is straightforward to apply airway remodeling effects in this simple model
representation, but limitations should be taken into account. The major errors
which may occur in this study are determined by the heterogeneity of the human
lung, i.e. inter-subject variability can affect the values from Table 1. However,
these values are reported in several studies [12, 13, 20, 23, 40] and they had
offered a good basis for investigations, originally measured from excised lungs
[26, 39] and then in plastic casts [31, 32]. One should recall that this study aims to
investigate the fractal-like geometry of the human lung, thus approximating it to
a dichotomous tree is necessary. Even though the airway-tree of the human lung
shows considerable irregularity, the principle of a systematic reduction of airway
size seems to apply [21]. It was later demonstrated by a systematic analysis that
the airway tree in different species shows a common fractal structure, in spite
of some gross differences in airway morphology [42]. On the other hand, it is
indeed interesting to quantify changes in the results if the degree of asymmetry
(which is not much discussed in literature) in the respiratory tree is taken into
account.
In terms of viscoelasticity, the elastic recoil of the lung is dominated at normal
breathing frequencies (around 0.25 Hz) by the nonlinear stress-strain characteristics of the healthy lung tissue [6, 35]. It is agreed that the main stress-bearing
constituents of lung tissue are collagen and elastin fibers. We have therefore
achieved the respective changes in the airway wall structure, by affecting the
balance between these two components.
5. Conclusions
In this paper, a mathematical model for the pressure and flow variations in
the respiratory tree has been developed based on similarity to the well-defined
Womersley theory, for oscillatory pressure and flow variations. The mathematical
model is developed for tidal breathing conditions, which reflect the breathing
at rest, allowing consequent assumptions to simplify the aero-dynamics (e.g.
laminar flow conditions). The influence of changes in radius and elastic modulus
with disease has been assessed in terms of mechanical parameters and total
impedance. In simulations for viscoelastic airways, we have found that variations
in these parameters correspond to the physiological insight and resemble the
measured data.
The parameters developed here can be further used to investigate cases of
pathologic subjects. The final aim is to obtain a direct relationship between
variations in model parameters and lung pathology, which would be of great
interest for clinical practice in follow-up studies and diagnosis. Additionally,
these changes can be correlated to variations in values of fractional order model
parameter, providing insight into the physiologic interpretation of such models.
Viscoelasticity and fractal structure in a model . . .
45
A first step in this direction would be to replace the classical Voigt model for
viscoelasticity by a non-integer order derivative model shortly presented here,
as well.
Appendix A. Transmission line equivalence
We consider the analogy to voltage as being the pressure p(x, t), and to
current as being the air-flow q(x, t), and we apply the transmission line theory.
Here x denotes the current position of the line. We shall make use of the complex
notation:
(A.1)
p(x, t) = P (x)ei(ωt−φP ) ,
q(x, t) = Q(x)ei(ωt−φQ )
where x is the longitudinal coordinate (m), t is the time (s), ω is the angular
√
frequency (rad/s), f is the frequency (Hz) and i is the complex unit i = −1.
The pressure and the flow are harmonics, with the modulus dependent solely on
the location within the transmission line (dx). The transmission line equations
link partial derivatives of p and g as follows:
∂p(x, t)
∂q
∂q(x, t)
∂p
= −rx q − lx ,
= −gx p − cx .
∂x
∂t
∂t
∂t
Introducing (A.1) in the first and second derivatives, gives, respectively:
(A.2)
∂P
= −(rx + iωlx )Q = −Zl Q,
∂x
∂Q
= −(gx + iωcx )P = −P/Zt ,
∂x
(A.3)
∂2P
∂Q
∂Q
= −Zl
,
= −(rx + iωlx )
∂x2
∂x
∂x
∂2Q
∂P
∂P
=−
/Zt ,
= −(gx + iωcx )
2
∂x
∂x
∂x
with Zl = rx +iωlx the longitudinal impedance, and Zt = 1/(gx +iωcx ) the transversal impedance. From (A.3) we obtain the system equations for P (x) and Q(x):
∂2P
∂2Q
−
Z
P/Z
=
0,
− Zl Q/Zt = 0.
t
l
∂x2
∂x2
p
p
Introducing the notation γ =
(rx + iωlx )(gx + jωcx ) =
Zl /Zt , it follows
that (A.4) can be rewritten as ∂ 2 P/∂x2 − γ 2 P = 0 and ∂ 2 Q/∂x2 − γ 2 Q = 0,
to which the solution is given by P (x) = Ae−γx + Be−γx and Q(x) =
Ce−γx + De−γx with complex coefficients A, B, C, D. Introducing this rela−γx − Be+γx ),
tion in (A.3),
p the system can be reduced
√ to Q(x) = 1/Z0 (Ae
with Z0 = (rx + iωlx )/(gx + iωcx ) = Zl Zt , in which Z0 is the characteristic
impedance of the transmission line cell.
(A.4)
46
C. M. Ionescu, W. Kosiński, R. De Keyser
Acknowledgement
Clara Ionescu acknowledges the financial support of the UGent-BOF grants
nr. 011/V0103 and 011/D04506.
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Received May 25, 2009; revised version December 17, 2009.